Determine how many solutions exist for the system of equations. ${3x+3y = -27}$ ${x+y = -9}$
Answer: Convert both equations to slope-intercept form: ${3x+3y = -27}$ $3x{-3x} + 3y = -27{-3x}$ $3y = -27-3x$ $y = -9-x$ ${y = -x-9}$ ${x+y = -9}$ $x{-x} + y = -9{-x}$ $y = -9-x$ ${y = -x-9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -x-9}$ ${y = -x-9}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${3x+3y = -27}$ is also a solution of ${x+y = -9}$, there are infinitely many solutions.